When you squeeze or pull on an object, you put stress on it. Imagine a rubber eraser that is being pressed or pulled. It is under stress.
The stress tensor is a way of describing the stress throughout the object precisely. The stress can vary from one point to another. For example, one side of the eraser might be pulled on (tensile stress) while the under is squeazed (compressive stress).
Normal Stress If normal stress is applied to a cube of elastic material, (and image for example that the stress is uniform throughout), the cube with stretch in the direction of the stress. (Figure) This stress could be applied in any direction or in more than one direction simultaneously. For a 3D object, it's sufficient to describe the normal stress with three numbers. That's a nubmer that tells how it's pulled along the x direction, the y direction, and the z direction.
Shear Stress Shear stress applied to a cube of elastic material (imagine that the stress is uniform throughout), it will shear. (Figure) You can use six numbers to represent the shear of an object. Look at how much face A moves relative to B in the x direction and the y direction. Look at how face C moves relative to D in the _ direction and the _ direction. Look at how face E moves relative to F in the _ direction and the _ direction. (Figure)
Because of symmetry only three numbers are needed to represent the shear stress.